Question

In a \(\triangle \mathrm{ABC}\), If $$ \left|\begin{array}{lll} 1 & \mathrm{a} & \mathrm{b} \\ 1 & \mathrm{c} & \mathrm{a} \\ 1 & \mathrm{~b} & \mathrm{c} \end{array}\right|=0 $$ then the value of \(64\left(\sin ^{2} \mathrm{~A}+\sin ^{2} \mathrm{~B}+\sin ^{2} \mathrm{C}\right)\) must be (a) 64 (b) 144 (c) 128 (d) 0

Short answer

The value of \(64(\sin^{2}A + \sin^{2}B + \sin^{2}C)\) is \(128\).

Step-by-step solution

Interpret the problem in terms of matrix determinant

The expression \[ \left|\begin{array}{ccc} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{array}\right| \] being equal to zero implies that the three rows of this matrix are linearly dependent. In other words, one row can be expressed as a combination of the other two.

Express one row as a combination of the other two

Let's express the third row as a sum of the first and the second rows. This gives us \(1\cdot R1 + 1\cdot R2 = R3\), which simplifies to \(a+b = b+c\). Hence, we find that \(a=c\).

Apply the Sine rule for triangles

In a triangle, by the sine rule, we know that \(a/\sin A = b/\sin B = c/\sin C\). As we obtained from step 2 that \(a=c\), we can infer that \(\sin A = \sin C\).

Use the identity \(\sin^{2}A + \sin^{2}B + \sin^{2}C = 2\)

In a triangle ABC, it is a known identity that \(\sin^{2}A + \sin^{2}B + \sin^{2}C = 2\). As \(\sin A = \sin C\), we can write the expression as \(\sin^{2}A + \sin^{2}B + \sin^{2}A = 2\).

Calculate the given expression

Our task is to find the value of \(64(\sin^{2}A + \sin^{2}B + \sin^{2}C)\). From step 4, replace \(\sin^{2}A + \sin^{2}B + \sin^{2}C\) with 2. Hence, the value of \(64(\sin^{2}A + \sin^{2}B + \sin^{2}C) = 64*2 = 128\). So the correct answer is (c) 128.

Key concepts

Matrix Determinant

Understanding matrix determinants is crucial for solving a wide range of mathematical problems, especially those involving linear algebra. In the context of triangles, a determinant can provide insights into the relationships between the sides. The determinant of a matrix is a special number that can tell us many things about the matrix, such as whether the matrix has an inverse, or if the system of linear equations associated with the matrix has a unique solution.

For a 3x3 matrix, the determinant is calculated using a specific formula involving the multiplication and subtraction of the elements in rows and columns. A determinant equal to zero, as in our problem, indicates that the rows of the matrix (or columns, depending on the perspective) are linearly dependent. This means that one of the rows can be expressed as a combination of the other two, signifying that there's no unique solution to the system, or geometrically, the vectors are coplanar in three-dimensional space.

Sine Rule in Triangles

The Sine Rule is a fundamental relationship in any triangle, providing an elegant link between its sides and angles. It states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle. Mathematically, it's expressed as \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).

In the given problem, applying the Sine Rule revealed that side lengths \(a\) and \(c\) are equal. Consequently, by the rule, their corresponding opposite angles, \(A\) and \(C\), have equal sines, leading to an understanding that \(\sin A = \sin C\). Knowing the Sine Rule and its implications on triangle properties is essential to solving problems related to the trigonometric ratios of angles and sides in triangles. It's a prime example of how trigonometry connects geometry with ratio and proportion.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are indispensable tools in solving problems across mathematics, from geometry to calculus. The identities often involve relationships between sine, cosine, and tangent functions.

An important identity used in the given problem is \(\sin^{2}A + \sin^{2}B + \sin^{2}C = 2\) for any triangle ABC. This is derived from the Pythagorean identity, which is itself a cornerstone of trigonometric identities, linking the square of sine and cosine functions for a given angle. Trigonometric identities simplify complex problems by transforming expressions into equivalent, often more manageable, forms. Understanding these identities and their applications can vastly improve problem-solving skills in trigonometry and beyond, as they reveal intrinsic properties of angles and their trigonometric functions.